Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
A detailed study of a quantum free particle on a pointed plane is presented in this paper. In particular, some questions posed in the very recent paper by M. A. Cirone et al, Phys. Rev. A 65, 022101 (2002) are clarified. Namely, the topological effects related to extracting a point from a plane are indicated. The proposed results are introduced concerning self-adjoint extensions of operators describing the free particle on a pointed plane as well as the role played by discrete symmetries in the analysis of such extensions.
We present an application of the affine covariant integral quantization (ACIQ) (Adv. Oper. Theory, 5, 2020; Adv. Oper. Theory, 7, 2022) to quantum mechanics on the punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), which comprises translations, rotations, and dilations of the plane. Due to the topology of the punctured plane, our quantization procedure gives rise to an affine vector potential. This potential can be interpreted as the Aharonov-Bohm (AB) gauge field produced by an infinite solenoid. This observation supports a reinterpretation of the AB effect: it emerges from the topological constraint imposed by the impenetrable coil rather than from an externally applied classical gauge field. In addition to this gauge structure, ACIQ also generates a repulsive, centrifugal-like scalar potential, a feature already encountered when applying ACIQ to motion on the half-line, whose phase space is the open half-plane. These results provide a new perspective on the AB effect, highlighting the central roles of topology and symmetry in quantum mechanics.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
In this paper, we apply covariant affine integral quantization to study motion in the 2D punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), representing transformations of the plane composed of translations, rotations, and dilations. Near the punctured point, we observe that the topology influencing quantum fluctuations gives rise to an affine vector potential, which can be interpreted as the Aharonov-Bohm (AB) gauge potential generated by an infinite coil. This finding suggests that the AB effect originates from the topology enforced by the impenetrable coil, rather than from a classical gauge potential. Our results offer a novel perspective on the AB effect, emphasizing the fundamental role of topology in quantum mechanics.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.